The lifespans of sloths in a particular zoo are normally distributed. The average sloth lives $18.9$ years; the standard deviation is $2$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a sloth living less than $16.9$ years.
Solution: $18.9$ $16.9$ $20.9$ $14.9$ $22.9$ $12.9$ $24.9$ $68\%$ $16\%$ $16\%$ We know the lifespans are normally distributed with an average lifespan of $18.9$ years. We know the standard deviation is $2$ years, so one standard deviation below the mean is $16.9$ years and one standard deviation above the mean is $20.9$ years. Two standard deviations below the mean is $14.9$ years and two standard deviations above the mean is $22.9$ years. Three standard deviations below the mean is $12.9$ years and three standard deviations above the mean is $24.9$ years. We are interested in the probability of a sloth living less than $16.9$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $68\%$ of the sloths will have lifespans within 1 standard deviation of the average lifespan. The remaining $32\%$ of the sloths will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({16\%})$ will live less than $16.9$ years and the other half $({16\%})$ will live longer than $20.9$ years. The probability of a particular sloth living less than $16.9$ years is ${16\%}$.